Problems 203 Substituting (12.27)into (12.18)and simplifying,the resulting constraint is generalized as follows: ck Bk I (12.29) k=1 Finally,substituting (12.28)into (12.29)and simplifying,one obtains: (12.30) Equation(12.30)represents the required relation between the overall and local (constituent)damage effect tensors. Problems Problem 12.1 Consider a composite system that consists of n constituents.In this case,the overall approach is schematically illustrated in Fig.12.3.In this case,derive (12.25)in detail. e A Fig.12.3.Schematic diagram illustrating the overall approach for composite ma- terials for Problem 12.1 Problem 12.2 Consider a composite system that consists of n constituents.In this case,the local approach is schematically illustrated in Fig.12.4.In this case,derive (12.25)in detail.Problems 203 Substituting (12.27) into (12.18) and simplifying, the resulting constraint is generalized as follows: n k=1 ckBk = I4 (12.29) Finally, substituting (12.28) into (12.29) and simplifying, one obtains: M = n k=1 ckMk−1 : B¯k −1 (12.30) Equation (12.30) represents the required relation between the overall and local (constituent) damage effect tensors. Problems Problem 12.1 Consider a composite system that consists of n constituents. In this case, the overall approach is schematically illustrated in Fig. 12.3. In this case, derive (12.25) in detail. Fig. 12.3. Schematic diagram illustrating the overall approach for composite ma￾terials for Problem 12.1 Problem 12.2 Consider a composite system that consists of n constituents. In this case, the local approach is schematically illustrated in Fig. 12.4. In this case, derive (12.25) in detail